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title: W14 NotesPolar coordinates are pairs of numbers
Polar coordinates have many redundancies: unlike Cartesian which are unique!
Polar coordinates cannot be added: they are not vector components!
⚠️ The transition formulas
Equations (as well as points) can also be converted to polar.
For
For
Compute the polar coordinates of
Solution
For
Next compute:
This angle is in Quadrant IV. We add
The radius is of course
For
Next compute:
This is the correct angle because Quadrant IV is SAFE. So the point in polar is
For example, let’s convert a shifted circle to polar. Say we have the Cartesian equation:
Then to find the polar we substitute
So this shifted circle is the polar graph of the polar function
To draw the polar graph of some function, it can help to first draw the Cartesian graph of the function. (In other words, set
This Cartesian graph may be called a graphing tool for the polar graph.
A limaçon is the polar graph of
Any limaçon shape can be obtained by adjusting
Limaçon satisfying
Limaçon satisfying
Limaçon satisfying
Limaçon satisfying
Limaçon satisfying
Limaçon satisfying
Transitions between limaçon types,
Notice the transition points at
The flat spot occurs when
The cusp occurs when
Roses are polar graphs of this form:
The pattern of petals:
The arclength of the polar graph of
To derive this formula, convert to Cartesian with parameter
From here you can apply the familiar arclength formula with
Let
Then:
Therefore:
Therefore:
Therefore:
Let us find the vertical tangents to the limaçon (the cardioid) given by
Consider the limaçon given by
Solution
The inner loop is traced by the moving point when
Therefore the length of the inner loop is given by this integral:
The “area under the curve” concept for graphs of functions in Cartesian coordinates translates to a “sectorial area” concept for polar graphs. To compute this area using an integral, we divide the region into Riemann sums of small sector slices.
To obtain a formula for the whole area, we need a formula for the area of each sector slice.
Let us verify that the area of a sector slice is
Take the angle
Then multiply this fraction by
Now use
One easily verifies this formula for a circle.
Let
The sectorial area between curves: